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Analysis of HDG Methods for Oseen Equations

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Abstract

We propose a hybridizable discontinuous Galerkin (HDG) method to numerically solve the Oseen equations which can be seen as the linearized version of the incompressible Navier-Stokes equations. We use same polynomial degree to approximate the velocity, its gradient and the pressure. With a special projection and postprocessing, we obtain optimal convergence for the velocity gradient and pressure and superconvergence for the velocity. Numerical results supporting our theoretical results are provided.

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Correspondence to Aycil Cesmelioglu.

Additional information

B. Cockburn supported in part by the National Science Foundation (Grant DMS-0712955).

N.C. Nguyen, J. Peraire supported in part by the Singapore-MIT Alliance.

Appendix: Approximation Properties of the Auxiliary Projection \(\varPi^{*}_{h}\)

Appendix: Approximation Properties of the Auxiliary Projection \(\varPi^{*}_{h}\)

The proofs in this appendix are quite similar to the proofs for the approximation properties of the projection Π h given in Sect. 3.3. We only need to recall that and that \(\widetilde{\varLambda}_{K}^{\max}\) is defined to be the maximum eigenvalue of \(\widetilde{\mathrm{S}}_{\boldsymbol{\beta}}\) over all faces of K.

1.1 A.1 Approximation Properties of Π ϕ

Proposition A.1

(Characterization of Π ϕ)

(A.1a)
(A.1b)

Proof

First equation follows from the definition of \(\varPi^{*}_{h}\). The second follows from (2.13d). Indeed if we take μP k (K) and using integration by parts,

Then, using integration by parts on the right hand side,

where the last equality holds by (2.13a) and (2.13c) and as . □

Let δ ϕ=Π ϕϕ k where ϕ k is the L 2-projection onto P k (K). Then, from (A.1a), δ ϕP k (K). Then, from the second characterization,

Now let μ=δ ϕ. Then, as , we have,

$$ \bigl\langle\mathrm{S}_{\boldsymbol{\beta}}\boldsymbol{\delta }^{\boldsymbol{\phi}},\boldsymbol{\delta}^{\boldsymbol{\phi}} \bigr\rangle_{\partial K}=b_{\boldsymbol{\phi}}\bigl(\boldsymbol {\delta}^{\boldsymbol{\phi}} \bigr)+b_{\varPhi }\bigl(\boldsymbol{\delta}^{\boldsymbol{\phi }}\bigr). $$

From Lemma 3.12,

$$\bigl\|\boldsymbol{\delta}^{\boldsymbol{\phi}}\bigr\|_{0,K}\leq Ch_K^{1/2} \bigl\|\boldsymbol{\delta}^{\boldsymbol{\phi}}\bigr\|_{0,F}. $$

Then, as S β is positive definite, for any face F of K,

$$\bigl\langle\mathrm{S}_{\boldsymbol{\beta}}\boldsymbol{\delta }^{\boldsymbol{\phi}},\boldsymbol{\delta}^{\boldsymbol{\phi}} \bigr\rangle_{\partial K}\geq\sum_F\sum _{i=1}^d \bigl\langle\mathrm{S}_{\boldsymbol{\beta}} \boldsymbol{\delta}^{\boldsymbol{\phi}},\boldsymbol{\delta }^{\boldsymbol{\phi}}\bigr\rangle_{F_i}\geq Ch_K^{-1} \varGamma_K^{\min }\bigl\| \boldsymbol{\delta}^{\boldsymbol{\phi}}\bigr\|_K^2. $$

Therefore,

$$\bigl\|\boldsymbol{\delta}^{\boldsymbol{\phi}}\bigr\|_K\leq C\frac {h_K}{\varGamma_K^{\min}} \bigl( \|b_{\boldsymbol{\phi} }\|+\|b_{\varPhi }\|\bigr). $$

b ϕ ∥ is bounded exactly as in Sect. 3.3, with one difference in the outcome. Rather than \(\varLambda_{K}^{\max}\), we have \(\widetilde{\varLambda}_{K}^{\max}\) and ∥b Φ ∥ is also bounded the same way except that we have \(|\nabla\cdot(\nu\varPhi +\phi\mathrm{I} )|_{k_{\sigma}}\) rather than \(|\nabla\cdot(\nu\varPhi -\phi\mathrm{I} )|_{k_{\sigma}}\).

1.2 A.2 Approximation Properties of νΠ Φ+Π ϕI

As in Sect. 3.3, we need two additional projections. We introduce a projection \(\widetilde{\mathrm{P}}^{1}\) similar to P1 as defined in Sect. 3.3 and we define \(\widetilde{\mathrm {P}}^{2}\) to suit to the form of the projection \(\varPi_{h}^{\star}\). Let \(\widetilde {\mathrm{P} }^{1}\varPhi \in\mathrm{P}_{k}(K)\) be such that

for all faces F of the simplex K except for an arbitrary one and let \(\widetilde{\mathrm{P}}^{2}\varPhi \in\mathrm{P}_{k}(K)\) be such that

for all faces F of the simplex K except for an arbitrary one.

Proposition A.2

(Characterization of νΠ Φ+Π ϕI)

(A.2a)
(A.2b)

for all faces F of K.

Proof

The first equation follows directly from (2.13a)–(2.13c). For the second pick an arbitrary face F of K and let wP k (F). Then, there exists μP k (K) such that μ=w on F. Therefore, in a similar fashion to the proof of the result for (ΠL−L)−(Πpp)I, splitting the integral over ∂K to F and ∂KF and using (2.13d) with this μ,

where

But by (A.1b) and integration by parts,

The first, third and the last terms vanish by cancellation and from the fact that wP k (K). The second, fourth and fifth terms vanish by (A.2a). □

The proof of the estimate for ν(Π ΦΦ) is very similar to the one for ν(ΠL−L). In short, from the representation of ν(Π ΦΦ) using (3.24), it boils down to bounding ν(Π ΦΦ)n F t for all faces F of K and for all . Using the projections we write . It is easy to show from the properties of \(\widetilde{\mathrm {P}}^{2}_{F}\), P F and (A.2b) that

Therefore,

The terms on the right hand side are bounded exactly the same way with the only difference being the equations defining . Now they are given by

for all faces F of K except for an arbitrarily chosen one.

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Cesmelioglu, A., Cockburn, B., Nguyen, N.C. et al. Analysis of HDG Methods for Oseen Equations. J Sci Comput 55, 392–431 (2013). https://doi.org/10.1007/s10915-012-9639-y

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  • DOI: https://doi.org/10.1007/s10915-012-9639-y

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